1. Semiparametric Regression Methodology for Censored Data This specific aim will be to develop methodology for the regression analysis of lifetime data under right, left and interval censoring, when one does not assume any particular parametric family of survival distributions. The multiple imputation approach presented in Tanner and Wong (1987a) will be used to develop one such approach, while techniques of Boos and Monahan (1983), Tanner and Wong (1984) and Wei and Tanner (1988) will be used to develop more faithful interpretations of the data augmentation algorithm (Tanner and Wong, 1987b) in the censored regression context. 2. Sampling From the Cox Partial Likelihood Wei and Tanner (1988) have developed a rejection/acceptance algorithm for sampling from the posterior (likelihood) in the case of parametric censored regression models. This specific aim is to extend this methodology to allow one to sample from the Cox partial likelihood and then to apply this methodology to a variety of problems related to the Cox model. In this way, the need to rely on asymptotic normal approximations for inferential purposes can be avoided. Problems include: parameter inference in small/unbalanced samples; inference relating to the survivor and cumulative hazard functions; missing (at random) covariates; and goodness-of-fit. 3. Sampling in the Context of Logistic Regression This specific aim is to investigate techniques to allow sampling from the likelihood for the logistic regression model and to then apply this methodology to a variety of problems related to the logistic regression model. In small samples or samples with covariate imbalance, the normal approximation to the scaled likelihood may not be appropriate. Problems to be addressed include: parameter inference in small/unbalanced samples; patching missing (at random) covariates; calculating the distribution of the "dose" corresponding to a given percentile; sequential design for estimating the percentile of a quantal response; and goodness-of-fit. 4. Development of Software to Accompany Methodology This specific aim is to develop transportable, documented, efficient and well-written code for the non-parametric estimation of the hazard function from grouped and censored data (Tanner and Wong, 1987a), as well as for the sampling from the posterior of Beta and Sigma in the normal, exponential, gamma and Weibull censored regression models (Wei and Tanner, 1988). Additional code will be written to support the methodology developed under this grant.